Plan
Antiderivative of a function and indefinite integral. Basic properties of the indefinite integral. Table of basic indefinite integrals. Basic integration methods: direct integration, substitution method, integration by parts.
Rational fractions. Integrating the simplest rational fractions. Integrating rational fractions.
Integration trigonometric functions. Integrating some irrational functions. Integrals that cannot be expressed through elementary functions.
Definite integral. Basic properties of a definite integral. Integral with variable upper limit. Newton-Leibniz formula. Basic methods for calculating a definite integral (change of variable, integration by parts).
Geometric Applications definite integral. Some applications of the definite integral in economics.
Improper integrals(integrals with infinite limits integrations, integrals of unbounded functions).
Antiderivative of a function and indefinite integral
In integral calculus, the main task is to find the function y=f(x) by its known derivative.
Definition 1. Function F(x) is called antiderivative functions f(x) on the interval ( a, b), if for any 
equality holds: 
or 
.
Theorem 1. Any continuous line on the interval [ a, b] function f(x) has an antiderivative on this segment F(x).
In what follows we will consider functions that are continuous on an interval.
Theorem 2. If the function F(x) is the antiderivative of the function f(x) on the interval ( a, b), then the set of all antiderivatives is given by the formula F(x)+WITH, Where WITH -constant number.
Proof.
Function F(x)+WITH is the antiderivative of the function f(x), because .
Let F(x) – another, different from F(x) antiderivative function f(x), i.e. 
. Then 
we have
which means that
,
Where WITH– constant number. Hence, 
Definition 2. The set of all antiderivative functions F(x)+WITH for function f(x) is called Not definite integral
from function f(x) and is indicated by the symbol 
.
Thus, by definition
(1)
In formula (1) f(x) is called integrand function, f(x)dx – integrand, x– integration variable, sign of the indefinite integral.
The operation of finding the indefinite integral of a function is called integration this function.
A geometrically indefinite integral is a family of curves 
(to each numerical value WITH corresponds to a certain curve of the family). The graph of each antiderivative (curve) is called integral curve. They do not intersect or touch each other. Only one integral curve passes through each point of the plane. All integral curves are obtained from one another parallel transfer along the axis Oh.
Basic properties of the indefinite integral
Let us consider the properties of the indefinite integral that follow from its definition.
1. The derivative of the indefinite integral is equal to the integrand, the differential of the indefinite integral is equal to integrand :
Proof.
Let Then
2. Indefinite integral of the differential of some function equal to the sum this function and an arbitrary constant:
Proof.
Really, .
3. Constant factor a() can be taken out as the sign of the indefinite integral:
4. Indefinite integral of an algebraic sum finite number functions is equal algebraic sum integrals of these functions:
5. If F(x) – antiderivative function f(x), That
Proof.
Really, 
6 (invariance of integration formulas). Any integration formula retains its form if integration variable replace by any differentiable function with this variable:
where u– differentiable function.
Table of basic indefinite integrals
Since integration is the inverse action of differentiation, most of the given formulas can be obtained by inverting corresponding formulas differentiation. In other words, the table basic formulas integration is obtained from the table of derivatives elementary functions when reading it backwards (from right to left).
Here is a table of the main indefinite integrals. (Note that here, as in differential calculus, the letter u can mean both the independent variable ( u=x), and a function of the independent variable ( u=u(x)).)
 |  |
||
 |  |
||
 |  |
||
 |  |
||
 |  |
||
 |  |
Integrals 1–12 are called tabular.
Some of the above formulas in the table of integrals, which do not have an analogue in the table of derivatives, are verified by differentiating their right-hand sides.
INTEGRAL CALCULUS- branch of mathematics in which integrals are studied various types, such as definite integral, indefinite integral, line integral, surface integral, double integral, triple integral etc., their properties, methods of calculation, as well as applications of these integrals to various problems of natural science.
The central formula of I. and. is the Newton-Leibniz formula (see Newton-Leibniz formula), connecting the definite and indefinite integrals (see Definite integral, Indefinite integral) functions - quantities defined in terms that are completely different from each other.
It is this formula that states that
under the following conditions and notations:
Line segment number axis, - continuous function, - partition of a segment by points , - segment , - point of a segment , 
, i.e., the maximum of the lengths of the segments , is an antiderivative function for , i.e., such that . The limit on the left side exists in the case continuous function, any method of refining the partition, in which , and any choice of points.
View limits 
arise when calculating many quantities associated with physical, geometric, etc. concepts. At the same time, calculating the antiderivative for simple functions It is carried out quite effectively according to the rules of I. and. These rules are based on the properties of differentiable functions studied in differential calculus, so that I. and. and differential calculus form an inseparable goal.
When moving from functions of one variable to functions of several variables, the content of information and. becomes much richer. The concepts of double, triple (and generally n-fold), superficial and curvilinear integrals. I. and. establishes the rules for calculating these integrals by reducing them to several times repeated calculations of certain integrals.
A separate section of I. and. functions of several variables is field theory (see Field theory), an essential part of which consists of theorems establishing the connection between integrals over a domain and integrals over the boundary of a domain (see Ostrogradsky formula, Green formula, Stokes formula).
In its further development I. and. led to the study of Stieltjes, Lebesgue, and Denjoy integrals, based on more general ideas than the integrals discussed above.
The emergence of I. and. associated with problems of calculating areas and volumes different bodies. Some advances in this direction took place back in Ancient Greece(Eudoxus of Kindsky, Archimedes, etc.). A revival of interest in problems of this kind took place in Europe in the 16th-17th centuries. By this time, European mathematicians had the opportunity to familiarize themselves with the works of Archimedes, translated into Latin language. But the main reason for such attention to And. and. appeared industrial development a number of European countries, which posed new challenges for mathematics. At this time, a great contribution to I. and. contributed by I. Kepler, B. Cavalieri, E. Torricelli, J. Wallis, B. Pascal, P. Fermat, X. Huygens.
Qualitative shift in I. and. the works of I. Newton and G. Leibniz appeared, who created a series common methods finding the limits of integral sums. Important had convenient symbolism I. and. (still used), introduced by G. Leibniz. After the works of I. Newton and G. Leibniz, many problems of artificial intelligence, which previously required significant skill for their solution, were reduced to a purely technical level. In this case, the differentiation formulas were especially important complex function, the rule for changing variables in definite and indefinite integrals, and (most of all) the Newton-Leibniz formula mentioned above.
Further historical development I. and. associated with the names of I. Bernoulli, L. Euler, O. Cauchy and Russian mathematicians M. V. Ostrogradsky, V. Ya. Bunyakovsky, P. L. Chebyshev.
I. and. together with differential calculus to this day it is one of the main mathematical tools of many physical and technical sciences.
Introduction
The integral symbol was introduced in 1675, and questions of integral calculus have been studied since 1696. Although the integral is studied mainly by mathematicians, physicists have also made their contribution to this science. Almost no physics formula can do without differential and integral calculus. Therefore, I decided to explore the integral and its application.
History of integral calculus
The history of the concept of integral is closely connected with problems of finding quadratures. Mathematicians of Ancient Greece and Rome called problems on the quadrature of one or another flat figure to calculate areas. Latin word quadratura is translated as “giving square shape" The need for a special term is explained by the fact that in ancient times (and later, until the 18th century) ideas about real numbers. Mathematicians operated with their geometric analogues or scalar quantities, which cannot be multiplied. Therefore, problems for finding areas had to be formulated, for example, like this: “Construct a square of equal area this circle" (This classical problem“On the squaring of a circle” of a circle” cannot, as is known, be solved with the help of a compass and a ruler.)
The symbol t was introduced by Leibniz (1675). This sign is a change Latin letter S (the first letter of the word summ a) The word integral itself was invented by J. Bernoulli (1690). It probably comes from the Latin integro, which translates as bringing to a previous state, restoring. (Indeed, the operation of integration “restores” the function by differentiating which the integrand was obtained.) Perhaps the origin of the term integral is different: the word integer means whole.
During the correspondence, I. Bernoulli and G. Leibniz agreed with J. Bernoulli’s proposal. At the same time, in 1696, the name of a new branch of mathematics appeared - integral calculus (calculus integralis), which was introduced by I. Bernoulli.
Other known terms related to integral calculus, appeared much later. The name “primitive function”, now in use, has replaced the earlier “primitive function”, which was introduced by Lagrange (1797). The Latin word primitivus is translated as “initial”: F(x) = m f(x)dx - initial (or original, or antiderivative) for f (x), which is obtained from F(x) by differentiation.
IN modern literature the set of all antiderivatives for the function f(x) is also called the indefinite integral. This concept was highlighted by Leibniz, who noted that everything antiderivative functions differ by an arbitrary constant b, they are called a definite integral (the designation was introduced by C. Fourier (1768-1830), but the limits of integration were already indicated by Euler).
Many significant achievements of the mathematicians of Ancient Greece in solving problems of finding quadratures (i.e. calculating areas) flat figures, as well as cubature (calculation of volumes) of bodies are associated with the use of the exhaustion method proposed by Eudoxus of Cnidus (c. 408 - c. 355 BC). Using this method, Eudoxus proved, for example, that the areas of two circles are related as the squares of their diameters, and the volume of a cone is equal to 1/3 of the volume of a cylinder having the same base and height.
Eudoxus' method was improved by Archimedes. The main stages characterizing Archimedes' method: 1) it is proved that the area of a circle less area any described about him regular polygon, But more area any inscribed; 2) it is proved that with an unlimited doubling of the number of sides, the difference in the areas of these polygons tends to zero; 3) to calculate the area of a circle, it remains to find the value to which the ratio of the area of a regular polygon tends when the number of its sides is unlimitedly doubled.
Using the exhaustion method and a number of other ingenious considerations (including the use of mechanics models), Archimedes solved many problems. He gave an estimate of the number p (3.10/71 Archimedes anticipated many of the ideas of integral calculus. (We add that in practice the first theorems on limits were proved by him.) But it took more than one and a half thousand years before these ideas found clear expression and were brought to the level of calculus. Mathematicians of the 17th century, who obtained many new results, learned from the works of Archimedes. Another method was also actively used - the method of indivisibles, which also originated in Ancient Greece (it is associated primarily with the atomistic views of Democritus). For example, they imagined a curved trapezoid (Fig. 1, a) to be composed of vertical segments of length f(x), to which, nevertheless, they assigned an area equal to the infinitesimal value f(x)dx. In accordance with this understanding, the required area was considered equal to the sum an infinitely large number of infinitely small areas. Sometimes it was even emphasized that the individual terms in this sum are zeros, but zeros of a special kind, which, added to an infinite number, give a well-defined positive sum. On such a now seemingly at least dubious basis, J. Kepler (1571-1630) in his writings “New Astronomy”. 1609 and “Stereometry of Wine Barrels” (1615) correctly calculated a number of areas (for example, the area of a figure bounded by an ellipse) and volumes (the body was cut into 6 finitely thin plates). These studies were continued by the Italian mathematicians B. Cavalieri (1598-1647) and E. Torricelli (1608-1647). The principle formulated by B. Cavalieri, introduced by him under some additional assumptions, retains its significance in our time. Let it be necessary to find the area of the figure shown in Figure 1, b, where the curves limiting the figure from above and below have the equations y = f(x) and y=f(x)+c. Imagining a figure made up of “indivisible”, in Cavalieri’s terminology, infinitely thin columns, we notice that they all have a total length c. By moving them in the vertical direction, we can form them into a rectangle with base b-a and height c. Therefore, the required area is equal to the area of the resulting rectangle, i.e. S = S1 = c (b - a). Cavalieri's general principle for the areas of plane figures is formulated as follows: Let the lines of a certain bundle of parallels intersect the figures Ф1 and Ф2 along segments of equal length (Fig. 1, c). Then the areas of the figures F1 and F2 are equal. A similar principle operates in stereometry and is useful in finding volumes. In the 17th century Many discoveries related to integral calculus were made. Thus, P. Fermat already in 1629 solved the problem of quadrature of any curve y = xn, where n is an integer (that is, he essentially derived the formula m xndx = (1/n+1)xn+1), and on this basis solved a series of problems to find centers of gravity. I. Kepler, when deducing his famous laws of planetary motion, actually relied on the idea of approximate integration. I. Barrow (1630-1677), Newton's teacher, came close to understanding the connection between integration and differentiation. Work on representing functions in the form of power series was of great importance. However, despite the significance of the results obtained by many extremely inventive mathematicians of the 17th century, calculus did not yet exist. It was necessary to highlight the general ideas underlying the solution of many particular problems, as well as to establish a connection between the operations of differentiation and integration, which gives a fairly general algorithm. This was done by Newton and Leibniz, who independently discovered a fact known as the Newton-Leibniz formula. Thus, the general method was finally formed. He still had to learn how to find antiderivatives of many functions, give new logical calculus, etc. But the main thing has already been done: differential and integral calculus has been created. Methods of mathematical analysis actively developed in the next century (first of all, the names of L. Euler, who completed a systematic study of the integration of elementary functions, and I. Bernoulli should be mentioned). Russian mathematicians M.V. took part in the development of integral calculus. Ostrogradsky (1801-1862), V.Ya. Bunyakovsky (1804-1889), P.L. Chebyshev (1821-1894). Of fundamental importance, in particular, were the results of Chebyshev, who proved that there are integrals that cannot be expressed through elementary functions. A rigorous presentation of the integral theory appeared only in the last century. The solution to this problem is associated with the names of O. Cauchy, one of the greatest mathematicians, the German scientist B. Riemann (1826-1866), and the French mathematician G. Darboux (1842-1917). Answers to many questions related to the existence of areas and volumes of figures were obtained with the creation of the theory of measure by C. Jordan (1838-1922). Various generalizations of the concept of integral already at the beginning of our century were proposed by the French mathematicians A. Lebesgue (1875-1941) and A. Denjoy (188 4-1974), the Soviet mathematician A.Ya. Khinchinchin (1894-1959). LIMIT THEOREMS OF PROBABILITY THEORY Chebyshev's inequality and its significance. Chebyshev's theorem. Bernoulli's theorem. The central limit theorem of probability theory (Lyapunov's theorem) and its use in mathematical statistics. Probability theory studies the patterns inherent in mass random phenomena. The limit theorems of probability theory establish the relationship between chance and necessity. The study of patterns manifested in mass random phenomena allows us to scientifically predict the results of future tests. Limit theorems of probability theory are divided into two groups, one of which is called law of large numbers, and the other - . This chapter discusses the following theorems related to the law of large numbers: Chebyshev's inequality, Chebyshev's theorems and Bernoulli's theorems. The law of large numbers consists of several theorems that prove the approximation of average characteristics, subject to certain conditions, to certain constant values. 1. Chebyshev’s inequality. If a random variable has a finite expectation and variance, then for any positive number the following inequality is true: that is, the probability that the deviation of a random variable from its mathematical expectation in absolute value will not exceed the difference between unity and the ratio of the variance of this random variable to the square. Let us now write down the probability of the event Chebyshev's inequality is valid for any distribution law of a random variable and applies to both positive and negative random variables. Inequality (9.2) limits from above the probability that a random variable will deviate from its mathematical expectation by an amount greater than. From this inequality it follows that as the dispersion decreases, the upper limit of the probability also decreases and the value of a random variable with a small dispersion is concentrated around its mathematical expectation. Example 1. To properly organize the assembly of a unit, it is necessary to estimate the probability with which the dimensions of the parts deviate from the middle of the tolerance field by no more than . It is known that the middle of the tolerance field coincides with the mathematical expectation of the dimensions of the machined parts, and the standard deviation is equal. Solution. According to the conditions of the problem we have: ,. In our case, the size of the parts being processed. Using Chebyshev's inequality, we obtain 2. Chebyshev's theorem. With a sufficiently large number of independent tests, it is possible, with a probability close to unity, to assert that the difference between the arithmetic mean of the observed values of a random variable and the mathematical expectation of this value in absolute value will be less than an arbitrarily small number, provided that the random variable has a finite dispersion, i.e. where is a positive number close to zero. Passing in curly brackets to the opposite event, we get Chebyshev's theorem allows one to judge the mathematical expectation using the arithmetic mean with sufficient accuracy, or vice versa: using the mathematical expectation to predict the expected value of the mean. Thus, on the basis of this theorem, it can be argued that if a sufficiently large number of measurements of a certain parameter are made with a device free from systematic error, then the arithmetic mean of the results of these measurements differs as little as possible from the true value of the measured parameter. Example 2. To determine the need for liquid metal and raw materials, the average weight of the casting of a liner for an automobile engine is determined selectively, since the weight of the casting calculated from the metal model differs from the actual weight. How many castings need to be taken so that with a probability greater than , it can be stated that the average weight of the selected castings differs from the calculated weight, accepted as the mathematical expectation, by no more than kg? It has been established that the standard deviation of weight is equal to kg
. Solution. According to the conditions of the problem we have ,, and taking into account equalities (4.4) and (4.5) - Substituting these problems here, we get where do we find it from? 3. Bernoulli's theorem. Bernoulli's theorem establishes a connection between the frequency of an event's occurrence and its probability. With a sufficiently large number of independent trials, it can be stated with a probability close to unity that the difference between the frequency of occurrence of an event in these trials and its probability in a separate trial in absolute value will be less than an arbitrarily small number, if the probability of the occurrence of this event in each trial is constant and equal to
. The statement of the theorem can be written as the following inequality: where and are any arbitrarily small positive numbers. Using the property of mathematical expectation and dispersion, as well as Chebyshev’s inequality, formula (9.3) can be written in the form When solving practical problems, it is sometimes necessary to estimate the probability of the greatest deviation of the frequency of occurrence of an event from its expected value. The random variable in this case is the number of occurrences of the event in independent trials. We have: Using Chebyshev’s inequality, in this case we obtain Example 3. Of the products sent to the assembly shop, randomly selected products were examined. Among them there were defective ones. Taking the proportion of defective products among those selected as the probability of producing a defective product, estimate the probability that the entire batch of defective products will contain no more than % and no less than %. Solution. Let us determine the probability of producing a defective product: The largest deviation of the frequency of occurrence of defective products from the probability in absolute value is equal to 4. Lyapunov's theorem. The considered theorems of the law of large numbers concern the issues of approximation of certain random variables to certain limiting values, regardless of their distribution law. In probability theory, there is another group of theorems concerning the limit laws of distribution of a sum of random variables. This group of theorems has the general name central limit theorem. The various forms of the central limit theorem differ from each other in the conditions imposed on the sum of the constituent random variables. The law of distribution of the sum of independent random variables ( 1) all quantities have finite mathematical expectations and variances: ; Where , 2) none of the quantities differs sharply in value from all the others: When solving many practical problems, the following formulation of Lyapunov’s theorem is used for the arithmetic mean of the observed values of the random variable, which is also a random variable (the conditions listed above are met): if a random variable has finite mathematical expectation and variance, then the distribution of the arithmetic mean Therefore, the probability of what is contained in the interval can be calculated using the formula Using the Laplace function (see Appendix 2), formula (9.5) can be written in the following form, convenient for calculations: It should be noted that the central limit theorem is valid not only for continuous, but also for discrete random variables. The practical significance of Lyapunov's theorem is enormous. Experience shows that the law of distribution of the sum of independent random variables comparable in their dispersion quickly approaches normal. Already with a number of terms of the order of ten, the law of distribution of the sum can be replaced by a normal one. A special case of the limit central theorem is Laplace’s theorem (see Chapter 3, paragraph 5). It considers the case when the random variables ,, are discrete, identically distributed and take only two possible values: and. For the application of this theorem in mathematical statistics, see paragraph 6 of Chapter 3. SELF-TEST QUESTIONS 1. What is called the law of large numbers? What is the meaning of this name? 2. Formulate Chebyshev’s inequality and Chebyshev’s theorem. 3. What is the role of limit theorems in probability theory? 4. Which of the laws of distribution appears as a limiting law? 5. What is Lyapunov’s central limit theorem? 6. How can Laplace's theorem be interpreted as a limit theorem in probability theory? TASKS FOR INDEPENDENT SOLUTION. 1. The length of manufactured products represents a random variable, the average value of which (mathematical expectation) is equal to cm. The variance of this quantity is . Using Chebyshev’s inequality, estimate the probability that: a) the deviation of the length of the manufactured product from its average value in absolute value will not exceed; b) the length of the product will be expressed by the number between and cm. Answer: a) ; b). 2. The device consists of independently operating elements. The probability of failure of each element over time is equal. Using Chebyshev’s inequality, estimate the probability that the absolute value of the difference between the number of failed elements and the average number (mathematical expectation) of failures over time will be less.
, (9.1)
, i.e. the event opposite to the event 
. It's obvious that
. (9.2)
.
, where is the average weight of the liner castings. If we apply Chebyshev’s inequality to a random variable, we obtain
,
.
,
, (9.3)
, (9.4)
,
.
.
.
; number of tests. Using formula (9.4), we find the required probability:
,
.
) approaches the normal distribution law with unlimited increase if the following conditions are met:
;
,
;
.
, calculated from the observed values of a random variable in independent tests, approaches the normal law with mathematical expectations of dispersion, i.e..
.
(9.5)
;
.
